Related papers: Dual flows in hyperbolic space and de Sitter space
A geometrical correspondence between maximal surfaces in anti-De Sitter space-time and minimal surfaces in the Riemannian product of the hyperbolic plane and the real line is established. New examples of maximal surfaces in anti-De Sitter…
In the framework laid down by Matveev and Shevchishin, superintegrability is achieved with one integral linear in the momenta (a Killing vector) and two extra integrals of of any degree above two in the momenta. However these extra…
In this paper we establish the short-time existence and uniqueness theorem for hyperbolic geometric flow, and prove the nonlinear stability of hyperbolic geometric flow defined on the Euclidean space with dimension larger than 4. Wave…
For a smooth, closed and uniformly $h$-convex hypersurface $M$ in $\mathbb{H}^{n+1}$, the horospherical Gauss map $G: M \rightarrow \mathbb{S}^n$ is a diffeomorphism. We consider the problem of finding a smooth, closed and uniformly…
There are three complete plane geometries of constant curvature: spherical, Euclidean and hyperbolic geometry. We explain how a closed oriented surface can carry a geometry which locally looks like one of these. Focussing on the hyperbolic…
We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant non-negative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a…
The paper studies a curvature flow linked to the physical phenomenon of wound closure. Under the flow we show that a closed, initially convex or close-to-convex curve shrinks to a round point in finite time. We also study the singularity,…
We construct new examples of immortal mean curvature flow of smooth embedded connected hypersurfaces in closed manifolds, which converge to minimal hypersurfaces with multiplicity $2$ as time approaches infinity.
We study rotational hypersurfaces with constant Gauss-Kronecker curvature. We solve the ODE for the generating curves of such hypersurfaces and analyze several geometric properties of such hypersurfaces. In particular, we discover a class…
In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under…
We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of…
The paper addresses the numerical approximation of two variants of hyperbolic mean curvature flow of surfaces in $\mathbb R^3$. For each evolution law we propose both a finite element method, as well as a finite difference scheme in the…
We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal…
The geometry of canal hypersurfaces of an n-dimensional conformal space C^n is studied. Such hypersurfaces are envelopes of r-parameter families of hyperspheres, 1 \leq r \leq n-2. In the present paper the conditions that characterize canal…
We consider compact hypersurfaces in an $(n+1)$-dimensional either Riemannian or Lorentzian space $N^{n+1}$ endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the…
We study high codimension mean curvature flow of a submanifold $\mathcal{M}^n$ of dimension $n$ in Euclidean space $\mathbb{R}^{n+k}$ subject to the quadratic curvature condition $ |A|^{2}\leq c_n |H|^{2}, c _n = \min\{ \frac{4}{3n} ,…
We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the…
We continue the study, initiated by the first two authors in \cite{IW19}, of Type-II curvature blow-up in mean curvature flow of complete noncompact embedded hypersurfaces. In particular, we construct mean curvature flow solutions, in the…
We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed…
In this paper, we consider the inverse hessian quotient curvature flow with star-shaped initial hypersurface in anti-de Sitter-Schwarzschild manifold. We prove that the solution exists for all time, and the second fundamental form converges…